Controlling high current (1000A) with MOSFETs

Look at page 4, fig.12, graph of safe operating area. That is exactly what you need.

You are talking about single pulse, right? You didn't mention any repetition or timing at all. If you open mosfet hard, say Rdson is 0.85mOhms. In case of 1000A the Vds will be less than 1V, so you have to look at the left side of graph.
There is no line for 100ms pulse, so you have to interpolate between DC and 10ms pulse. The safe current is much lower than 1000A. It is like 400A. And it is the maximum.

it depends of on/off ratio, how much heat is produced. These transistors blocks have one limitation, that is a heat transfer. They are not so good when cooling down, another drawback is the large gate capacitance, so you will need a very expensive and powerful gate driver, even more if you will put them in parallel.

IMO you can do a better circuit if you use a bunch of D2Pak transistors in parallel. D2Pak can handle more current, but then you would need some complicated PCB.

You should worry a bit more about the super capacitors. Some Murata "high current" models are rated for up to 10A. Other super capacitors have rating in the milliampere range.